Thermal phenomena associated with water transport across a fuel cell membrane: Soret and Dufour effects

Journal of Membrane Science 431 (2013) 96–104

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Thermal phenomena associated with water transport across a fuel cell membrane: Soret and Dufour effects K. Glavatskiy a,n, J.G. Pharoah b, S. Kjelstrup a a b

Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway Queen’s University, Kingston, ON, Canada, K7L 3N6

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 August 2012 Received in revised form 17 December 2012 Accepted 19 December 2012 Available online 31 December 2012

We present calculations of the coupling effects that take place when heat and water are transported across a membrane relevant to fuel cells, using the theory of non-equilibrium thermodynamics. Numerical results are given for the Nafion membrane bounded by surfaces of molecular thickness in contact with water vapor of varying relative humidity. Analytical expressions for thermal effects of water transport are given. We show how reversible heat transport (Dufour effects) can be understood in terms of coupling coefficients (heats of transfers). The sign of the enthalpy of adsorption of water in the membrane determines the sign of the coupling coefficient, the Dufour and Soret effect as well as thermal osmosis effects meaning that the effect can be large at interfaces. We show how data presented in the literature can be understood in terms of the presented theory. Using common estimates for transport properties in the membrane and its surface, we find that the more detailed equations predict a 10–30% variation in the heat and mass fluxes as the membrane thickness drops below 1 mm. Analysis of experiments on thermal osmosis suggests that more accurate measurements on the water content as a function of activity are required. & 2012 Elsevier B.V. All rights reserved.

Keywords: Non-equilibrium thermodynamics Heat fluxes Water flux Boundary conditions Thermal osmosis Membrane transport

1. Introduction It is well established in the literature on membrane transport processes, for instance thanks to the work of Tasaka and coworkers, that a transmembrane temperature difference is able to drive a mass flux [1,2]. The phenomenon is called thermal osmosis, or thermal diffusion, first described by Soret, see [3,4]. Vice versa, a thermal flux is also affected by mass transport (the Dufour effect). These reciprocal effects can be systematically described by the theory of non-equilibrium thermodynamics [1,3,4]. So far an overall approach was taken in the analysis of the experiments, but a more detailed treatment of the interface region is now possible [5]. It has been increasingly clear that the membrane interface adjacent to the bulk phase of liquid or vapor can pose a separate resistance to water and heat transport [6–10]. This can be ascribed to the variation in intensive variables across the interface. A rapid enthalpy drop may for instance introduce an excess resistance to heat and mass transfer. It has been documented that this is so for water transport [9,10]. However, the model used by Monroe et al. [9] and Romero and Merida [10] does not consider the possible coupling of the water flux to the heat flux. Their

n

Corresponding author. Tel.: þ47 47244779. E-mail addresses: [email protected], [email protected] (K. Glavatskiy). 0376-7388/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.memsci.2012.12.023

model thus is not able to predict thermal osmosis. In this work, we shall generalize the work of Monroe et al. [9], Romero and Merida [10], and Tasaka et al. [1,2] and address the membraneinterface system in such a way that all effects are taken into account in a systematic manner. It is well known from studies of this cell that temperature gradients exists across the membrane [11–13], and that the water transport is large (up to 104 kg=m2 s) and essential to good cell performance [14]. The water content, which depends on the temperature [15], varies from 3–4 to 15 water molecules per ionic site [15,16]. The chemical structure of the membrane can also affect the membrane water content, which may lead to the so-called Schroeder’s paradox, when the membrane water content is different for a membrane exposed to liquid water and to water vapor even though the chemical potentials of these phases are the same if in equilibrium [17, p. 42]. Water transport due to a transmembrane temperature difference was measured for fuel cell membrane materials by Kim and coworkers, see [18,19] and references therein. The authors documented by experiment the significance of the thermal driving force. Fluxes directed along the temperature gradient were defined as positive thermal diffusion, while fluxes in the opposite direction were defined as negative. The magnitude and sign varied with the layered materials and their stacking. The origin and sign of the effect was discussed [20]. In spite of the extensive research on the fuel cell performance, all aspects are not clear. For instance, Kim and Mench emphasize that there is no consensus on the measured water flux between

K. Glavatskiy et al. / Journal of Membrane Science 431 (2013) 96–104

different groups [18]. This may be caused by unaccounted effects of the interface, as we shall see. As a first step in a systematic investigation of the role of the Soret and Dufour effects in the ion exchange membrane, including also the membrane interfaces as separate systems in the analysis, we shall study coupled the flux of heat and water across the membrane with its interfaces. We shall relax the assumption of equilibrium at the phase boundary. This assumption is not special for fuel cell membranes, but applies to most phase transitions [21]. The study may therefore also be of interest to other membrane processes, like pressure retarded osmosis, pervaporation, or separation. One reason why the assumption of equilibrium at the interface may fail is that relatively large heat effects can be associated with membrane adsorption of water. For a Nafion membrane the heat adsorption of water in the acid membrane varies with water content [22]. For dry membranes values above 80 kJ/mol have been observed [22]. This motivated our wish to study fuel cell related systems. The systematic treatment of an interface (or surface) in terms of non-equilibrium thermodynamics [5] offers a possibility to circumvent the common assumption of zero driving forces (equilibrium) at the membrane interface. This work aims to investigate the importance of this possibility for a membrane system that is relevant to the polymer electrolyte fuel cell. It is common to distinguish between thermo-osmosis, thermodiffusion and other phenomena related to the water flux due to the temperature difference [13]. Non-equilibrium thermodynamics shows that all these phenomena can be described in a systematic way using coupling of heat and mass transfer. In the electroneutral membrane system there exist only two driving forces: the difference in the temperature and the difference in the chemical potential of water. The chemical potential of water depends on the chemical structure of the surroundings, water pressure and concentration. Coupling means that the water flux can be caused by a temperature difference, and that the heat flux can be caused by the chemical potential difference alone. The driving force can be expressed by the difference in partial entropy in the fluid and membrane phase [23], the difference in the pressure, or alternatively the difference in the concentration. However, care must be taken to give the same entropy production. Non-equilibrium thermodynamics allows one to treat all these phenomena in a systematic manner. The key quantity which appears in this description is the heat of transfer, the ratio between the heat flux and the water flux when the temperature difference is zero. According to the Onsager symmetry in coefficients in force–flux relations, this quantity is also responsible for the water flux due to the temperature gradient. It is the aim of the present work to solve equations for heat and water transport across a Nafion membrane bounded by two vapor phases under various relevant fuel cell conditions. Some typical relevant conditions will be studied to gain insight into isothermal heat transport, thermal osmosis and water transport at large. The aim is to give a new basis for analysis of experimental results where heat and mass transport occurs. We shall also compare the analysis to common descriptions using common values for transport coefficients as far as possible, and see that the extended description is necessary under several conditions. We start by giving the transport equations, as derived from the entropy production for heat and mass transport across a Nafion membrane with vapor boundaries [5,24]. Several equivalent sets of equations are available, and we provide two; one for convenient calculations of variable profiles, and the other for the description of experiments. The first set is solved for stationary state conditions, using data available in the literature and some best estimates. The second set is used to analyze experimental

97

and computational results. Cases are presented in the results section that have physical relevance for the membrane as a fuel cell membrane.

2. System Consider the heterogeneous system illustrated in Fig. 1 consisting of a membrane m and two adjacent surfaces, sa and sc. The surfaces (interfaces) separate the membrane from a vapor at a and c. The width of the interfaces is much smaller than the width of the membrane. The purpose of the figure is to illustrate that we consider the interfacial layers to be as important as the membrane layer itself. One can choose to deal with relatively thin surfaces as being two dimensions, while thicker layers can be treated as homogeneous (three dimensions) systems [5]. There is transport of heat and water through this heterogeneous system. The entropy production, as determined in nonequilibrium thermodynamics, defines the relevant fluxes and forces of transport for each relevant thermodynamic system. In the present case there are three systems; the three-dimensional membrane and its 2 two-dimensional interfaces. We do not go into much detail about that, but refer to the original literature [5] for more explanations and terminology. We shall consider stationary state only. Furthermore, we consider all the properties to be dependent only on the coordinate x across the membrane. The water flux and the energy flux through the membrane are constant in this case. The water flux, Jm, and the energy flux, Jq, satisfy the conservation equations for mass and energy; @r=@t ¼ dJ m =dx and @u=@t ¼ dJq =dx, where r is the density (in kg/m3) and u is the internal energy density (in J/m3). We distinguish now between the incoming ‘‘i’’ and outgoing ‘‘o’’ sides of each layer in Fig. 1. The sensible or measurable heat flux J0q at the outgoing side of the surface, J0A,o (or J0C,o q q ), differs from the heat flux on the incoming side of the surface, J 0A,i (or J 0C,i q q ), but they are related by the definition of the energy flux [5,24]: Jq  J 0q þ HJ m , where H is the partial specific enthalpy of water. This energy flux depends on a reference, the standard enthalpy of the formation of water. We aim to find the measurable heat flux, J0q , which can be related directly to experiments. We take advantage of the constant nature of the energy flux in the calculations and find: 0A,i A,i A,o Þ ¼ J m Dvap HA J0A,o q J q ¼ J m ðH H

ð1Þ

0C,o J0C,i ¼ Jm ðHC,o HC,i Þ ¼ Jm Dvap HC q J q

ð2Þ

Here we introduced Dvap H, the enthalpy of vaporization for water from a condensed state inside the membrane to a vapor state outside. The enthalpy differences across the different surfaces sa and sc may differ in magnitude if the conditions at a and c differ. This difference creates a discontinuity in the measurable heat flux at the boundary.

Fig. 1. The schematic representation of the system, showing the membrane, m, and its surfaces, sa and sc. The width of the layers is not to scale and is given only for illustration. For further notation, see the text.

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K. Glavatskiy et al. / Journal of Membrane Science 431 (2013) 96–104

The discontinuity in the heat flux at the interfaces leads to jumps in the temperature Dio T and the chemical potential Dio m across each of the interfaces. The common approach is to assume equilibrium at the surface, meaning that the jumps of these quantities between points A,o and A,i, as well as between C,i and C,o are zero. In the present study, this assumption shall be relaxed. Conditions for validity of the assumptions shall be examined. A complete non-equilibrium thermodynamic description requires the expression for the entropy production, the amount of entropy produced in the non-equilibrium system per unit of time and surface area, s. The two equivalent formulations which are relevant are linked as follows:

DmT

1 T

s ¼ J0q D Jm

T

1 m ¼ Jq D Jm D T T

ð3Þ

Here D means the difference in a quantity across a layer: right minus left. Furthermore, DmT is the chemical potential difference evaluated at the temperature of the right (left) side for a heat flux evaluated at the left (right) side, while Dðm=TÞ is the difference in the Planck potential, with m being the chemical potential. The subtle difference between the expressions were described in detail in the literature [5,24]. Here, we observe that the expression to the right of the first equality sign gives fluxes and corresponding driving forces which can be experimentally determined, while the expression behind the second equality sign is suitable for integration across the membrane, when the fluxes are constant. Each set in Eq. (3) determines relations between the fluxes and forces. We give the first set of such relations below, the second set in the Calculations section, and the interrelation between the two sets, due to their equivalence, in the Appendix. For the first set of variables Jm and J 0q , belonging to the first layer to the left in Fig. 1 [24,5], we have J 0A,o ¼ wsa q Jm ¼ 

Dio T sa

d

þ qn,sa Jm

 n,sa  Dsa q Dio T Dio mT sa þ sa A,o @m=@c9T T d d sa

ð4Þ

Here d is the thickness of the layer, and wsa , Dsa and qn,sa are the surface thermal conductivity at stationary state, diffusion coefficient of water and the water heat of transfer, respectively. The heat of transfer is defined as the ratio between the heat flux and the water flux, when the temperature difference is zero. This coefficient expresses the coupling of heat and mass transfer and has so far mostly been neglected in fuel cell modeling. In the expression for the heat flux, it can be used to quantify the Dufour effect. While the value of qn,sa is small for homogeneous phases, it becomes large at surfaces [5]. The magnitude of the water flux is determined by the magnitude of the diffusion coefficient. Similar equations are also valid for differences between the points A,o and C,i, as well as C,i and C,o, i.e. across the membrane and the right hand side surface. Instead of the interfacial propersa ties wsa , Dsa , qn,sa and d , one has to use the corresponding membrane conductance wm , diffusion coefficient Dm , heat of m transfer qn,m and thickness d . Corresponding values apply for the last layer, but we shall take the properties of the two interface layers to be the same. The transport properties may depend on the temperature and other local properties, but we shall take them as constant for temperatures around 350 K. The analytical solution for temperatures around 350 K shows that the error introduced by temperature variation in the transport properties is rather small. This is expected, since the overall temperature jumps are rather small. When the heat of transfer qn ¼ 0, by definition there is no coupling between heat and mass flux. In this case the heat flux is caused only by the temperature difference, and is given by the

first term in the first of Eq. (4), which is Fourier’s law. Furthermore, the water flux is caused only by the chemical potential difference at constant temperature Dio mT and is given by the second term in the second of Eq. (4), which is Fick’s law. The chemical potential difference Dio mT can be expressed in terms of either the pressure difference Dio p or the water content difference Dio l, depending on whether we are outside or inside the membrane. Both quantities can be described by the activity, see below. The expression for the water flux has two terms. We recognize the last term as related to Fick’s law at isothermal conditions. The sign of the heat of transfer determines the sign of thermal diffusion or thermal osmosis. In the case of a negligible temperature difference, we speak of reversible heat transport as a consequence of the heat of transfer. In this case, the Fourier term is negligible. Still, in order to obey the energy balance at the boundaries, the heat (enthalpy) transported per mole to the interface may be substantial. The energy balance gives the heat of transfer its physical interpretation as the fraction of the enthalpy for the phase change contributed by the layer to which the heat flux belongs. The first equation of Eq. (4) is reduced to Fourier’s law in the absence of a mass flux. The magnitude of the correction to Fourier’s law depends on the heat of transfer, which is rather small in the membrane qn,m (order of magnitude 10 J/mol [25]), but is expected to be substantial for the surface due to a large value of the enthalpy of vaporization [5].

3. The solution procedure In the calculations, we use the total heat flux Jq rather than the measurable heat flux J 0q . The advantage of using the total heat flux and the mass flux as variables in the solution procedure is that analytical solutions can be found for the profiles of the driving forces across the membrane. From these we can reconstruct the parameters of interest to the experimentalist. Furthermore, we shall choose to use resistances (of the whole layer) rather than (specific local) conductivities. The advantage of using resistances is that all layers are placed in series, which makes the resistance of the whole system simply a sum of resistances of separate layers. With this choice of variables, the flux equations from Eq. (3) take the following form for each layer:

D

1 ¼ r qq J q þr qm J m T

D

m T

¼ r mq J q þ r mm J m

ð5Þ

with resistance coefficients r. According to Onsager reciprocal relations, r qm ¼ r mq . The resistances r are related to the measurable quantities in the following way (see Appendix A): r qq ¼

d

wT 2

r qm ¼ ðqn þHÞ r mm ¼ ðqn þHÞ2

d

wT 2 d

wT 2

þ

d @m TD @r T

ð6Þ

where again d is the thickness of the layer, H is the partial specific enthalpy of water in the layer and @m=@r9T is the derivative of the chemical potential with respect to the water density at constant temperature. As the reference for the enthalpy we use the standard enthalpy of formation. Other choices are also possible, as it is not possible to relate this solution to measurements. The equations relating measurable properties are independent of the choice.

K. Glavatskiy et al. / Journal of Membrane Science 431 (2013) 96–104

When these equations are applied to an interfacial region, we use the values of the temperature and partial specific enthalpy on the vapor side of the surface, T A,i and HA,i (T C,o and HC,o ), while in the membrane region we use the average temperature and partial specific enthalpy of the membrane, T m and Hm . Each of the three above regions are characterized by the three sets of the coefficients r m , r sa and r sc . Writing down Eq. (5) for each layer, summing them up and taking into account that the fluxes Jq and Jm are constant, we get the following equation: 1 T C,o 



mC,o T C,o

1 T A,i þ

T A,i

¼ Rmq J q þRmm J m

1

y

sa

ð7Þ

where the resistances Rik (where indices i and k can both be either q or m) across the whole system are m sc Rik ¼ r sa ik þ r ik þ r ik

ð8Þ

In a typical experiment we control the boundary conditions, such as the temperatures T A,i and T C,o , or the water vapor pressures. The vapor pressures are related to the chemical potentials mA,i and mC,o . Furthermore, we can control the water flux Jm. Thus, knowing the values of the resistances Rik, one can solve the system. Using the thermodynamic relations, we obtain

D

m T

¼

DmT T

þ HD

1 T

ð9Þ

where DmT is the difference between the corresponding chemical potentials at the same temperature. This quantity is convenient to use, since it is directly related to the measurable properties such as the pressure, while Dðm=TÞ is not. This expression is a good approximation for reasonably small values of the temperature difference. In our modeling the largest temperature drop will be 10 K, which we consider to be small enough. Instead of the chemical potential it is convenient to use the activity a, which is a measurable quantity and is defined as

mða,TÞ ¼ mn ðTÞ þ RT ln aðTÞ

The thermal conductivity w and the diffusion coefficient D appear always in combination with the layer thickness, in d=w and d=D, respectively. These combinations are measures of the main resistance to heat and mass transfer of the region of thickness d. In the absence of much information on the interfacial resistances, we model the main interface resistances as fractions of the corresponding membrane resistance. Thus, interfacial conductivities wsa and Dsa (or wsc and Dsc respectively) are modeled in the following way:

wsa ¼ wsc ¼ wm

¼ Rqq Jq þ Rqm J m

mA,i

99

ð10Þ

where m ðTÞ is the chemical potential of the saturated water. In fuel cell modeling it is common to define the activity as a  p=pn ðTÞ, where pn ðTÞ is the saturation pressure [16]. The use of the pressure instead of fugacity here indicates that we consider the vapor as the ideal gas. This is a poor approximation, when vapor becomes very condensed, however is common for modeling. Combining Eqs. (9) and (10), we obtain an expression for the jump of Plank’s potential used in Eq. (7)   mC,o mA,i 1 1 pC,o  A,i ¼ H C,o  A,i þ R ln A,i ð11Þ C,o p T T T T n

where pA,i and pC,o are the pressures of the vapor on the sa and sc surfaces, which can be controlled experimentally. The main aim of this analysis is to show that the surface plays an important role for the profiles of the temperature and the chemical potential across the system. The reason for this is that the values of the resistances rsa and rsc are non-negligible, meaning that the surface influences the properties of the system significantly. According to Eq. (6), these resistances are expressions of the thermal conductivity, diffusion coefficient and the heat of transfer for the interface. Such quantities are mostly not available in the literature so we model them with reference to the membrane values (see below). One attempt to estimate experimentally the interfacial resistance was done [9]. In our modeling we also separate the interface contributions from the membrane contribution, and study their relative impact on the total fluxes. Consider the transport coefficients in Eq. (4). The equations will be similar for the membrane and the other interface.

sc

D ¼D ¼

1

d

Dm

ð12Þ

where y and d are the thermal conductivity scaling factor and the diffusional scaling factor, respectively. Their meaning is how much the interfacial resistance is different from the bulk resistance of the same thickness as the interface. Therefore, if y ¼ 1 (or d ¼ 1), the interfacial layer has the same resistance as the corresponding part of the membrane and therefore no excess resistance. Larger values of y (or d) indicate that the interface has excess resistance. According to our experience [6–8] surface indeed has the excess resistance and therefore we should consider the values of these coefficients greater than one. Monroe et al. [9] argue that the interface has a similar resistance to mass transfer per unit of thickness as the membrane, meaning that the scaling factor for mass transfer is close to unity. For an interface model with series of layers, it was shown [5, Chapter 8] that the heat of transfer was proportional to the difference between the partial enthalpy on the right and the left side of the surface. This gives a convenient formula, in the absence of better information: qn,sa ¼ kDvap H

ð13Þ

where k is the heat of transfer scaling factor with a value between 0 and 1, and the enthalpy of vaporization was defined above. The value given by kinetic theory of gases is 0.2 for the liquid–vapor transition [5]. The enthalpy difference will, in a majority of cases, determine the sign of the thermal osmosis effect. The same expression is used for qn,sc . Furthermore, the sign of k is decisive for the sign of the water flux, which we will not discuss in this paper.

4. Input data The equations were solved using thermodynamic data and transport data for the polymer fuel cell, see e.g. [26], plus some typical boundary conditions. In the above expressions a number of material properties that depend on the temperature and the activity are needed. We assume now that the jumps are so small that the profiles inside the surface and membrane are linear. This gives also a rationale for using the average temperature of the membrane T m ¼ ðT A,i þ T C,o Þ=2 and average water activity in the membrane am  ðpA,i =pn ðT A,i Þ þ pC,o =pn ðT C,o ÞÞ=2 to find the coefficients. 4.1. Transport data The membrane thickness was set to 180 mm (Nafion 117), while the interface thickness was set to 18 nm to indicate that it is of molecular dimensions. The value is somewhat arbitrary, and must be seen in connection with the factors y and d which we shall vary by orders of magnitude. Membrane transport data. A common value for the membrane diffusion coefficient is 1010 m2 =s [16]. The membrane thermal

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K. Glavatskiy et al. / Journal of Membrane Science 431 (2013) 96–104

conductivity was 0.2 W/(K m) [27], and the membrane heat of transfer was estimated to be  15 J/mol using the data from [25]. Surface transport data. One of the aims of this paper is to find the effect of the interface. Therefore, we need to estimate the interfacial transport properties. Transfer coefficients for the surface were estimated with the help of scaling factors. According to Eq. (12), the specific thermal conductivity of the surface layer is ws ¼ wm =y. Our task is therefore by varying the value of the thermal conductivity scaling factor to adjust the results of the model with the results of the experiments. The same arguments apply for the interfacial diffusion coefficient and the diffusional scaling factor d. The sign of the heat of transfer depends on the nature of the water interactions with the membrane: it is different for hydrophobic and hydrophilic membranes. The experiments by Kim et al. [18,19] can be understood in this context. The chemical structure of Nafion has both hydrophobic and hydrophilic sites. According to Reucroft et al. [22] the exothermal effect dominates, meaning that the enthalpy difference is negative, and the sign of k is positive, and in condensation of a liquid. We use the value of k¼0.2 in Eq. (13) in accordance with kinetic theory. Experiments of James and Phillips [28] support this value. They report a heat of transfer for water at an interface with sulfuric acid near  8 kJ/ mol, which is near 20% of the enthalpy of condensation. 4.2. Enthalpies The partial specific enthalpy of the water in the membrane and vapor may differ significantly. The enthalpy of water in each of the surface layers was taken from the standard enthalpies of formation for the vapor phase at T  ¼ 298 K: ðTT  Þ Hs ðTÞ ¼ Df Hvapor þcvapor p vapor

ð14Þ cvapor p

with Df H ¼ 13,435 kJ/kg and ¼ 1996 J=ðK kgÞ. The enthalpy of water in the membrane phase differs from the enthalpy of water in the gas phase by the enthalpy of vaporization Hm ðTÞ ¼ Hs ðTÞDvap H

ð15Þ

where the enthalpy of vaporization depends on the water content in the membrane l, or the number of water molecules per sulfonic site [22,29]:

Dvap H ¼ 88:76:7l, l o 6:5 Dvap H ¼ 45, l 46:5

ð16Þ

where the numbers have dimension kJ/mol. When the water content in the vapor exceeds that of saturation, the enthalpy of adsorption becomes constant and equal to the value of evaporation for pure water. 4.3. Membrane water activity The water content of a Nafion membrane is l  Mc=r, where M is the polymer molecular weight (is equal to 1.1 kg/mol), r is the membrane density (is equal to 1.64 kg/m3) and c is the water concentration. The water activity is defined by Eq. (10). In the membrane phase the relationship between the water activity and the water content was determined by Springer et al. [16]:

l ¼ 0:043þ 17:81a39:85a2 þ36:0a3 , 0 oa o1 l ¼ 14 þ1:4ða1Þ, 1 ra o3

ð17Þ

The relation is widely used, and recommended in Handbooks [26]. From the relation we find the derivative of the chemical potential with respect to concentration at constant temperature, which is used in the expression for r m mm in Eq. (6)

@mm RT M ¼ @r T ra @l=@a MH2 O

ð18Þ

where R¼8.31 J/K mol is the gas constant and M H2 O ¼ 0:018 kg=mol is the water molar mass. In the surface layer we assume the water to be the ideal gas and use a ¼ cRT=pn ðTÞ. This gives for the derivative of the chemical potential in the surface @msa ðRTÞ2 ¼ ð19Þ @r T pM H2 O The saturation pressure of water is well known, see e.g. [30]: pn ðTÞ ¼ 2846:4 þ 411:24ðT273:15Þ10:554ðT273:15Þ2 þ 0:16636ðT273:15Þ3 ;

ð20Þ n

From these expressions we can calculate l whenever p and p are known along with the temperature. This is the case at the A,i and C,o sides of the surface. But it is also possible, knowing the temperature and the chemical potential in the membrane, to find a value of the activity which can be translated into the water content using Eq. (17).

5. Results and discussion We report the solution to the general set of equations to the particular problem of thermal osmosis. Thermal osmosis produces a net water flux from a temperature difference ðT A,i aT C,o Þ, with zero chemical driving force across the membrane ðpA,i ¼ pC,o  pÞ. We use the pressure equal to 3  104 Pa, while the temperature difference is varied from 0 K up to 10 K. The temperature on the left a-side is always larger than the temperature on the right c-side. Furthermore, the positive value of the water flux corresponds to the water flow from left to right, i.e. from the hot to the cold side. The negative value of the water flux corresponds to the water flow from the cold to the hot side, respectively. The water flux can be found from Eq. (7) using Eq. (11):     Rqq Rqm 1 1 pC,o Jm ¼  þH  A,i þR ln A,i ð21Þ 2 C,o Rqq p T Rqq Rmm Rqm T

5.1. Interfacial parameters We first investigate the dependence of the water flux on the parameters of the interface, namely the heat conductivity scaling factor y and the diffusion scaling factor d. Fig. 2 shows the dependence of the water flux on the heat conductivity scaling factor for the interface for different temperatures. We can see that the magnitude of the flux varies both with the temperature difference and the interfacial resistance. The calculated flux does not reach the value presented in [18], which is of the order of 103 kg=m2 s. However, it agrees with the results in [25,31]. The high value of the thermal resistance makes the water flux smaller. However in the wide range of interfacial thermal resistances the value of the water flux does not change. Fig. 3 shows the dependence of the water flux on the diffusion coefficient scaling factor for the interface. We can see that the magnitude of the water flux depends significantly on the magnitude of the interfacial diffusion coefficient. Small interfacial permeability leads to small values of the water flux, while large interfacial permeability leads to the large values of the water flux. Again, we can see an agreement on the magnitude of the flux with the results of [25,31], but not of [18]. To investigate the dependence on y and d further, we plot twodimensional figures where both of them can vary. Fig. 4 shows the landscape of the water flux when the temperature difference

K. Glavatskiy et al. / Journal of Membrane Science 431 (2013) 96–104

−5 4 x 10

3.5 104

3 2.5

3 2.5

103

2

θ

− J / kg/(m2 s)

− J / kg/(m2x10S−5)

105

Δ T = 10 K ΔT=5K ΔT=1K

3.5

101

2

102

1.5

1.5

1

1

101 0.5

0.5 0 100

100 100

101

102

103

104

θ Fig. 2. Dependence of the water flux on the heat conductivity scaling factor for the interface for different values of the temperature difference across a Nafion membrane of thickness 180 mm. The boundary conditions are T A,i ¼ T C,o þ DT, T C,o ¼ 350 K, pA,i ¼ pC,o ¼ 104 Pa, while d ¼ 1, k¼ 0.2.

103

104

105

Fig. 4. Dependence of the water flux on d and y. The membrane and boundary conditions are as above, T A,i ¼ 360 K, T C,o ¼ 350 K, pA,i ¼ pC,o ¼ 104 Pa, k¼0.2.

−4 1.2 x 10

Δ T = 10 K ΔT=5K ΔT=1K

1 − J / mol/(m2 s)

− J / kg/(m2 s)

102 δ

10−4

10−6

10−8

10−10 0 10

101

105

0.8 0.6 0.4 0.2

Δ T = 10 K ΔT=5K ΔT=1K

101

0 102

103

104

0

50

105

δ Fig. 3. Dependence of the water flux on the diffusion coefficient scaling factor for the interface for different values of the temperature difference across a Nafion membrane of thickness 180 mm. The boundary conditions are T A,i ¼ T C,o þ DT, T C,o ¼ 350 K, pA,i ¼ pC,o ¼ 104 Pa, while y ¼ 1, k¼ 0.2.

across the membrane is 10 K. We see from Fig. 4 that the magnitude of J is still determined mostly by the value of d, but not much on the value of y. Comparing the order of magnitude of the flux with the experimental values, we may estimate that for Nafion membrane used in the experiments (k ¼0.2) we should use the following values of the interfacial scaling factors: y  100 and d  10. This would mean that we impose an excess resistivity to both heat and mass transfer to the surface. 5.2. Boundary conditions We now consider how the water flux depend on boundary conditions. The results are given in Figs. 5 and 6. In particular, Fig. 5 shows the dependence of the water flux on the membrane thickness. It is well known from the experiments that its magnitude increases when the membrane thickness decreases, which is indeed the case.

100 dm / μm

150

200

Fig. 5. Dependence of the water flux on the membrane thickness. The boundary conditions are T A,i ¼ 360 K, T C,o ¼ 350 K, pA,i ¼ pC,o ¼ 104 Pa.

Furthermore, Fig. 6 shows the dependence of the water flux on the temperature difference across the membrane. We also see that the flux is proportional to the temperature difference.

5.3. Thermostat properties Next, we considered the dependence of the water flux on the temperature on the a-side of the membrane and the gas pressure. Fig. 7 shows the corresponding profiles if we use the data for the water content l taken from [16] as given in Eq. (17). A nonmonotonous and discontinuous dependence of the water flux on the temperature and pressure is observed. This is clearly unphysical. Jumps in the values of flux happen, because the value of the water content changes from one region of the functional dependence to another in Eq. (17). The change notifies the point where the gas becomes saturated and starts to condense. There is a jump in the derivative of the water content with respect to the activity, which leads to the jump of the water flux. It is therefore beneficial to smoothen the water content behavior as a function of the activity. To illustrate this in more

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detail, consider as an example Eq. (22) rather than Eq. (17) 2

3

l ¼ 0:043þ 17:81a39:85a þ36:0a , 0 oa o1 l ¼ 16:82:8 expðða1Þ=0:0607Þ, 1r a o 3

ð22Þ

making dl=da continuous at the point a ¼1, going towards the value 16.8 when a goes to 3, as in [16]. The corresponding profiles of l and dl=da are given in Fig. 8. Using such a dependence to obtain the dependence of the water flux on the temperature on the a-side of the membrane and the gas pressure, we find the results plotted in Fig. 9. Sharp peaks in these profiles apparently correspond to the discontinuity in the second derivative of lðaÞ. Furthermore, the non-monotonous behavior apparently corresponds to the two different regimes in the water content behavior as a function of the water activity: before and after saturation. Clearly the example does also not capture the correct physics. A precise measurement of the dependence of a on l may give a smoother curve.

6. Conclusions and perspectives We have studied water and heat transport across a Nafion membrane under conditions relevant for the fuel cell. The set

of equations were derived from the consistent non-equilibrium thermodynamic procedure. The use of non-equilibrium thermodynamics allows one to treat the coupling of the water flux due to the temperature gradient in a systematic manner, and defines the driving forces, differences in terms of pressure (Eq. (11)), water content (Eq. (10) together with Eq. (17)) or partial entropy [23]. The model includes the interface between the membrane and the fluid as an additional resistance, motivated by data in the literature [9]. We see that such layers can affect the overall performance of the cell in major ways. By comparing with the available experimental data, we have estimated the adjustable parameters used to model the interface. With these values of parameters we are able to obtain the typical dependencies of the water flux on the boundary conditions. We have shown that the interface plays an important role in the fuel cell performance. A variation of the interfacial permeability changes the magnitude of the water flux a lot. We have shown that it is of major importance to account for coupling coefficients in the description of the heat and mass fluxes across the membrane and its bounding surfaces. The water and heat fluxes depend heavily on these coefficients. Accurate predictions of water and heat fluxes in fuel cells require therefore a model compatible with that derived in non-equilibrium thermodynamics.

−5 2.5 x 10

50

λ dλ/da

2

40

1.5

30

λ, dλ/da

− J / kg/(m2 s)

TA = 360 K, k = 0.2 TA = 350 K, k = 0.2

1

0.5

0

20

10

0

2

4

6

8

0

10

0

0.5

TA − TC / K

1

1.5

a

Fig. 6. Dependence of the water flux on the temperature difference across the membrane. T A,i ¼ 360 K, while T C,o ¼ T A,i DT. pA,i ¼ pC,o ¼ 104 Pa.

Fig. 8. Dependence of the water content on the water activity according to Eq. (22).

−5 2.5 x 10

−5 3.5 x 10

3 2 − J / kg/(m2 s)

− J / kg/(m2 s)

2.5 1.5

1

2 1.5 1

0.5 0.5 0 320

330

340 TA / K

350

360

0 104

105 p / Pa

Fig. 7. Dependence of the water flux on the membrane temperature and pressure. (a) T A,i varies from 320 K to 360 K, while T C,o is 10 K or 5 K lower. pA,i ¼ pC,o ¼ 104 Pa. (b) pA,i ¼ pC,o ¼ p, while T A,i ¼ T C,o þ DT, T C,o ¼ 350 K.

K. Glavatskiy et al. / Journal of Membrane Science 431 (2013) 96–104

−5 2.5 x 10

103

−5 3.5 x 10

3 2 − J / kg/(m2 s)

− J / kg/(m2 s)

2.5 1.5

1

2 1.5 1

0.5 0.5 0 320

330

340

350

0 104

360

105

TA / K

p / Pa

Fig. 9. Dependence of the water flux on the membrane temperature and pressure. (a) T A,i varies from 320 K to 360 K, while T C,o is 10 K or 5 K lower. pA,i ¼ pC,o ¼ 104 Pa. (b) pA,i ¼ pC,o ¼ p, while T A,i ¼ T C,o þ DT, T C,o ¼ 350 K.

The typical heat conductivity and permeability of the interface may be rather large, meaning that the interface does not rise a barrier to the water and heat flux. The variation of the interfacial heat conductivity does, however, not affect much the magnitude of the water flux. The model and the results presented above can in this manner be used to interpret experimental results. We have seen that experiments on thermal osmosis [25], when discussed in this broader context, indicate that the mechanism of water transport is not comparable to diffusion, but contains additional contributions. Experiments are frequently referred to the overall membrane conditions. In order to be able to decompose the overall results, to give results specific for the membrane or the surfaces, more detailed theoretical expressions are needed. The present set of equations are complete, obey the Onsager symmetry conditions, and can be used to arrive at information on the leading and possibly negligible terms. This opens a possibility to understand the discrepancy between the different results observed in the experiments by the different values of the parameters for the interface. In particular, the controversy on the sign of the water flux [18] may be related to the value of the coefficient k in Eq. (13). We have found a non-monotonous behavior of the water flux as a function of the reservoir properties, such as pressure and temperature. This behavior is explained by the different regimes of the water content dependence on the water activity. Furthermore, this can explain the discrepancy between the measurement of different groups, as they may have different reservoir conditions. A precise measurement of the water content dependence on the water activity may provide a better understanding of the difference in reported measurements.

Acknowledgments SK and JP are grateful to the Centre of Advanced Study at the Norwegian Academy of Science and Letters. All authors are grateful for support from the Norwegian Research Council, grant no. 167336/V30. Appendix A. Transport coefficients In order to relate the coefficients r w ik , where the subscripts i and k can both be either q or m and w denotes one of the regions m,

or sa, or sc, to the measurable transport coefficients we do the following. The entropy production sw for each of the region can be written in the following way: 1 T

sw ¼ Jq D Jm Dw

m

ðA:1Þ

T

This gives the following force–flux relations:

Dw

1 w ¼ rw qq J q þ r qm J m T

Dw

m T

w ¼ rw mq J q þ r mm J m

ðA:2Þ

which is the same as Eq. (4). While the energy flux Jq is constant, it is not directly measurable quantity. In contrast, the heat flux J0q  J q HJm , where H is the partial enthalpy of the component, is measurable. If one uses this flux in the expression for the entropy production, one gets 1 T

sw ¼ J0q D Jm

Dw mT T

ðA:3Þ

where Dw mT  TðDw ðm=TÞHDw ð1=TÞÞ is the chemical potential difference taken at constant temperature. The corresponding force–flux relations take the following form:

Dw 

1 0 w ¼ rw qq J q þ rqm J m T

Dw m T

0 w ¼ rw mq J q þ rmm J m

ðA:4Þ

where the coefficients rik are related to rik in the following way: w rw qq ¼ rqq w w rw mq ¼ rmq H rqq w w rw qm ¼ rqm H rqq 2 w w w w rw mm ¼ rmm Hðrqm þ rmq Þ þ H rqq

ðA:5Þ

This can be easily verified by comparing Eq. (A.2) with Eq. (A.4). According to [5], the coefficients rik are related to the thermal conductivity w, diffusion coefficient D and the heat of transfer qn as

rwqq ¼

1

wT 2

rwmq =d ¼ rwqm =d ¼  n 2

rwmm =d ¼

ðq Þ

wT 2

þ

qn

wT 2 @m=@c9T TD

ðA:6Þ

104

K. Glavatskiy et al. / Journal of Membrane Science 431 (2013) 96–104

where d is the thickness of the region. Here we took into account that relations in [5] are given for the specific resistivities, while we use the resistances of the whole region. The latter ones divided by the thickness of the region give the former ones. Substituting Eq. (A.6) into Eq. (A.5) we get Eq. (6). References [1] M. Tasaka, T. Mizuta, O. Sekiguchi, Mass transfer through a polymer membrane due to a temperature gradient, J. Membrane Sci. 54 (1990) 191–204. [2] T. Suzuki, K. Iwano, R. Kiyono, M. Tasaka, Thermoosmosis and transported entropy of water across hydrocarbonsulfonic acid-type cation-exchange membranes, Bull. Chem. Soc. Jpn. 68 (1995) 493–501. [3] A. Katchalsky, P. Curran, Nonequilibrium Thermodynamics in Biophysics, Harvard University Press, Cambridge, Massachusetts, 1975. [4] K.S. Førland, T. Førland, S. Kjelstrup Ratkje, Irreversible Thermodynamics. Theory and Application, 1st ed., Wiley, Chichester, 1988. [5] S. Kjelstrup, D. Bedeaux, Non-equilibrium Thermodynamics of Heterogeneous Systems, Series on Advances in Statistical Mechanics, vol. 16, World Scientific, Singapore, 2008. [6] A. Røsjorde, D. Bedeaux, S. Kjelstrup, B. Hafskjold, Non-equilibrium molecular dynamics simulations of steady-state heat and mass transport in condensation II: transfer coefficients, J. Colloid Interface Sci. 240 (2001) 355–364. [7] I. Inzoli, S. Kjelstrup, D. Bedeaux, J.M. Simon, Transfer coefficients for the liquid–vapor interface of a two-component mixture, Chem. Eng. Sci. 66 (2010) 4533–4548. [8] K. Glavatskiy, D. Bedeaux, Resistances for heat and mass transfer through a liquid–vapor interface in a binary mixture, J. Chem. Phys. 133 (2010) 234501. [9] Charles W. Monroe, Tatiana Romero, Walter Merida, Michael Eikerling, A vaporization-exchange model for water sorption and flux in Nafion, J. Membrane Sci. 324 (2008) 1–6. [10] Tatiana Romero, Walter Merida, Water transport in liquid and vapour equilibrated NafionTM membranes, J. Membrane Sci. 338 (2009) 135–144. [11] P.J.S. Vie, S. Kjelstrup, Thermal conductivities from temperature profiles in the polymer electrolyte fuel cell, Electrochim. Acta 49 (2004) 1069–1077. [12] J.G. Pharoah, O.S. Burheim, On the temperature distribution in polymer electrolyte fuel cells, J. Power Sources 195 (2010) 5235–5245. [13] A. Thomas, G. Maranzana, S. Didierjean, J. Dillet, O. Lottin, Thermal effect on water transport in proton exchange membrane fuel cell, Fuel Cells 12 (2012) 212–224. [14] A.Z. Weber, J. Newman, Effects of microporous layers in polymer electrolyte fuel cells, Chem. Rev. 104 (2005) 4679–4726.

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